# Candida Bowtell

**About me**

I am currently a Research Fellow at the University of Warwick, working with Richard Montgomery. Prior to that I was a Research Fellow in Combinatorics at the University of Birmingham. I completed my DPhil (PhD) at the University of Oxford under the supervision of Peter Keevash.

My research interests lie in extremal and probabilistic combinatorics, with a particular focus on problems in extremal (hyper)graph theory.

## Papers

**The n-queens problem**** **(with Peter Keevash).

We consider the number of ways to place n queens on an n x n chessboard so that no two can attack one another. We also consider the toroidal variant, proving lower bounds for both. These problems date back to 1848 and 1918, with Gauss and Pólya among the many interested in these problems. Writing T(n) for the number of ways to place n non-attacking queens on the toroidal n x n chessboard and Q(n) for the number of ways to place n non-attacking queens on the non-toroidal n x n board, Pólya showed that T(n)>0 if and only if n is not divisible by 2 or 3, and subsequently Luria showed that T(n) \leq ((1+o(1))n/e^3)^n for all n, and conjectured equality when n is not divisible by 2 or 3. We show that T(n)=((1+o(1))n/e^3)^n for all n not divisible by 2 or 3, and Q(n) \geq ((1+o(1))n/e^3)^n for all natural numbers n, confirming the conjecture of Luria as well as a conjecture of Rivin, Vardi and Zimmerman concerning both the toroidal and classical problems. We translate the toroidal problem to a problem of counting perfect matchings in a hypergraph, and combine a random greedy counting process with an absorbing strategy that develops ideas from the methods of randomised algebraic construction and iterative absorption.

**Matchings in extremal k-partite k-graphs** (with Richard Mycroft).

We consider the largest size of a matching guaranteed in a k-partite k-graph with given minimum partite codegree conditions. That is, rather than a minimum codegree condition, for each (k-1)-tuple with a vertex from each part except part i, the condition is that this tuple is in at least a_i edges. Hang, Zang and Zhao showed that for every ε >0 and sufficiently large n, with a_1, a_2 >= εn, H contains a matching of size at least \min\{n-1, \sum_{i \in [k]} a_i\}, answering and generalising a question of Rödl and Ruciński. However, their methods fail when all of a_2,..., a_k are small. We consider the remaining cases and are able to show when \sum_{i=2}^k a_i <= \sqrt{n/(k+1)}, H in fact contains a matching of size at least \min\{n, \sum_{i \in [k]} a_i\} and, in the instances that remain, that almost all graphs also satisfy \min\{n-1, \sum_{i \in [k]} a_i\}. Our proof uses a novel approach, making use of Aharoni and Haxell's `Hall's theorem for hypergraphs' and rainbow matchings.

**A degree sequence strengthening of the vertex degree threshold for a perfect matching in 3-uniform hypergraphs** (with Joseph Hyde), *SIAM Journal on Discrete Mathematics, accepted*.

The asymptotic minimum vertex degree threshold for a perfect matching in in 3-uniform hypergraphs is known to be (5/9+o(1))n^2/2. We improve on this result, giving infinitely many degree sequences in which a third of vertices can have degree (4/9+o(1))n^2/2 or below, and a perfect matching is still guaranteed.

**Classification of Maximum Hittings by Large Families** (with Richard Mycroft), *Graphs and Combinatorics* **36, **27–39 (2020).

For integers r and n, where n is sufficiently large, and for every set X ⊆ [n] we determine the maximal left-compressed intersecting families A ⊆ [n]^r which achieve maximum hitting with X (i.e. have the most members which intersect X). This answers a question of Barber, who extended previous results by Borg to characterise those sets X for which maximum hitting is achieved by the star.

## Talks

*The n-queens problem:*

Combinatorics Seminar, University of Warwick, October 2022.

Combinatorial Mathematics Society of Australasia Seminar, September 2022 (online).

Symposium Discrete Mathematics, Hamburg University of Technology, September 2022.

12th WOMBL 1-day meeting in additive combinatorics and analytic number theory, King's College London, September 2022.

RSA, Gniezno, August 2022.

BCC, Extremal combinatorics mini-symposium, University of Lancaster, July 2022.

Second Armenian Workshop on Graphs, Combinatorics, and Probability, July 2022.

7th Lake Michigan Workshop on Combinatorics and Graph Theory, University of Illinois at Chicago, May 2022.

ACiD Seminar, Durham University, February 2022.

Seminar on Combinatorics, Games and Optimisation, LSE, January 2022.

EPC Webinar, December 2021 (online).

Nachmittagsseminar der Gruppe Diskrete Mathematik, TU Ilmenau, December 2021 (online).

Combinatorics Seminar, University of Birmingham, December 2021 (hybrid).

Combinatorics Seminar, University of Oxford, November 2021.

Combinatorics Seminar, UCL, November 2021.

Discrete Seminar, Umeå University, November 2021 (online).

Physics and Probability Seminar, Harvard CMSA, October 2021 (online).

*Chess puzzles: from recreational maths to fundamental mathematical structures, *North meets South Colloquium, University of Oxford, December 2021 (hybrid).

*Matchings in k-partite k-graphs:*

SIAM - DM21, Extremal graph theory and combinatorics mini-symposium, Spokane, Washington, July 2021 (online).

BCC, Durham University, July 2021 (online).

Combinatorics Seminar, University of Bristol, December 2020 (online).

*Maximum hittings by maximal left-compressed intersecting families*:

DIMAP Seminar, University of Warwick, November 2019.

BCC, University of Birmingham, August 2019.

RSA, Zurich, July 2019.

25th Postgraduate Combinatorial Conference, London School of Economics, June 2018.

*Intersecting set systems*, Combinatorics Seminar, University of Birmingham, February 2015.

## Teaching

At the University of Oxford I was involved with several maths courses for the department and a subset of the colleges, both teaching and marking work.

2020-2021:

Class and consultation tutor for 4

^{th}Year Probabilistic CombinatoricsClass tutor for 3

^{rd}Year Graph TheoryCollege and revision tutor for 2

^{nd}Year Probability

2019-2020:

Class and consultation tutor for 3

^{rd}Year Graph TheoryClass tutor for 4

^{th}Year Probabilistic CombinatoricsCollege tutor for 2

^{nd}Year Graph Theory

2018-2019:

College tutor for 1

^{st}Year Linear Algebra and AnalysisCollege and revision tutor for 2

^{nd}Year Graph Theory

2017-2018:

College and revision tutor for 1

^{st}Year ProbabilityTA for 3

^{rd}Year Graph TheoryTA for 4

^{th}Year Probabilistic Combinatorics